QUEUES WITH INTERRUPTION IN MARKOVIAN ENVIRONMENT

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. This paper carries out an analysis of an M/M/1 queue with service interruption, where the interruptions are caused by different environmental factors which are the states of a Markov chain. Interruption due to only one factor is allowed at a time and during an interruption period, no more interruption is allowed. The server is unaware of the interruption until a random amount of time elapses. The duration of unidentified interruption is assessed with the help of the random clock that started ticking at the onset of the interruption. The interruption is identified only when the random clock is realized. A new service is commenced to the interrupted customer after fixing the interruption. The kind of service to be started after fixing the interruption depends on the environmental factor that caused the interruption. Even though any number of interruptions can occur during the service of a customer, the maximum number of interruptions is restricted to a finite number K and if the number of interruptions exceeds this maximum, the customer goes out of the system without completing the service. The superclock measures the total duration of interruption in the service of a customer. This clock starts ticking at the onset of the first interruption to a customer in service. When interruption is fixed, this clock stops. If another interruption strikes the same customer’s service, the superclock starts from the earlier position where it stopped ticking and so on. Finally, when the customer leaves the system the superclock is reset to the zero position. On realization of superclock the customer goes out of the system without completing service. We analysed service process to obtain stability condition. Then we performed steady state analysis and important performance measures are obtained.


INTRODUCTION
We come across different types of interruptions in all walks of life like internet browsing, banking, medical emergency, etc. Queue with interruption plays an important role in the fast growing world of network communication. The interruption can be due to different factors such as server breakdown, preemptive priority, vacation, etc. The pioneers of the concept of the queue with interruption are White and Christie [1]. The arrival rate during the breakdown of the server and service rate after the breakdown of the server could be different from what it was when the server was working. Bhaskar Sen Gupta [2] analysed a model with the above characteristics.
A survey paper by Krishnamoorthy et.al [3] gives a detailed description of the research on queueing models with service interruptions induced by break down as well as those with customer induced service break. Verghese et.al [4] introduced another form of interruption called customer induced interruption. Here the self interrupted customer enters into a finite buffer.
On finding the buffer full the interrupted customer goes out of the system. On completion of their interruption, the customers in the buffer enter another buffer. They are given preemptive priority over new customers.
In almost all papers on queues with service interruptions, the service is either resumed or repeated on the removal of interruption. In the resumption of service, the interrupted service is resumed from the point of interruption, and in the case of a restart a new service begins after fixing the interruption. Keilson [5] considers a queueing model with repeat or resumption of the interrupted service. In [6] Krishnamoorthy and co authors studied a queueing model with interruption where the special stress was on deciding whether to repeat or resume an interrupted service. A paper by Krishnamoorthy, Deepak and Pramod [7] considers a queueing model with service interruption and repair where the decision on, whether to repeat or resume the interrupted service is following the realization of a phase type distributed random clock.
In [8] Krishnamoorthy et.al studied a queueing model with interruption where the interruption is caused by n random environmental factors and the maximum number of interruptions that can happen to a customer is restricted to a finite number. If the interruption exceeds that predetermined number the customer goes out of the system without completing service. The server is unaware of the interruption for a random duration and that duration is determined with the help of a random clock started at the onset of the interruption. The total duration of interruption to a particular customer is determined with the help of a super clock started at the onset of the first interruption to a customer in service and which is reset to zero position when the customer in service leaves the system either by completing or by not completing the service. In this paper we consider a queueing system in which interruption is caused by different environmental factors which are the states of a Markov chain, ie. the environmental factor causing the previous interruption influences the environmental factor causing the next interruption. All other assumptions are the same as in [8].
The remaining discussion on the model is arranged as follows. Section 2 gives a detailed description of the model. In section 3 the service process with interruption is analysed. Section 4 is developed based on the number of interruptions. Detailed analysis of the queueing model is included in section 5. Section 6 is devoted to important performance measures. A numerical illustration is included in section 7.

MODEL DESCRIPTION
In this queueing system arrivals occur according to a Poisson process with parameter λ . The service time is exponentially distributed with parameter µ. During service, interruption occurs according to a Poisson process with parameter β . There are n environmental factors causing interruption which form a Markov chain with initial probability vector p i ; i = 1, 2, . . . , n and transition probability matrix P = (p i j ); i, j = 1, 2, . . . , n. The fixing time is exponentially distributed with parameter α i , only if the i th factor is the cause of the interruption. On fixing the interruption, a new service starts which is exponentially distributed with parameter µ i , provided the i th factor caused the interruption.
The super clock, which is Erlang distributed with parameter γ and a, started at the beginning of the first interruption that strikes the customer in service, will freeze when the new service starts after interruption and it will again start functioning from the position where it stopped when another interruption occurs. Sometimes the super clock may realize before the random clock. The number of interruptions during the service of a customer is limited to K. On realization of the super clock or when the number of interruptions exceeds K, whichever occurs first to the customer in service, he goes out of the system without completing service.

ANALYSIS OF SERVICE PROCESS WITH INTERRUPTIONS
a Markov Chain with 3naK + 1 transient states and one absorbing state. The absorbing state represents the customer moving out of the system, either after service completion or without completing service. Here S(t) denotes the status of the server at time t: 3, if service is continuing after interruption at time t; I 1 (t) denotes the number of interruptions occurred until time t to the current customer in service. I 1 (t) varies from 0 to K. I 2 (t) corresponds to the environmental factor that caused the current interruption to the service. In this model we consider n environmental factors.
where e is a column vector with all its entries equal to 1 and of appropriate order.
In matrix T , T ii < 0, 1 ≤ i ≤ 3nak + 1 and T i j ≥ 0 for i = j. Also Te + T 0 = 0. The initial probability vector is ζ = (1, 0, . . .). Here the service process with interruption follows PH Distribution. So using the property that residual service time in a phase type distributed service process is also phase type we see that • The Probability distribution F (.) of X with initial probability vector ζ is given by • Probability for service completion without any interruption is, P(s) = ζ (−C 0 ) −1 C 03 .
• Probability for customer leaving the system due to realization of super clock, • Probability for customer leaving the system due to occurrence of maximum number of .
Lemma: The expected time for service completion/customer leaving the system without completing service due to realization of super clock or attaining maximum K is E(ST ) = ζ (-T) −1 e, and hence the expected service rate is µ s = 1/E(ST ).

EXPECTED NUMBER OF INTERRUPTIONS
The expected number of interruption during the service of a customer can be calculated by . D 0 = C 0 , The initial probability vector ζ = 1 0 0 . . .
From the above matrices we get the following system characteristics: • Probability for absorption after r(r ≤ K) interruption is given by • Expected number of interruptions for a customer, E(I) = ∑ K r=1 rN r . • Probability for service completion after r(1 ≤ r ≤ K) interruptions, • Expected number of interruptions before service completion for a single service Having computed the measures indicated above, we describe the queueing model and the condition for it to be stable. Its infinitesimal generator Q is given by

Consider the queueing model
Theorem: The system Z is stable when λ < µ s . elements. We assume that x 2 = x 1 .R, and where R is the minimal non-negative solution to the matrix quadratic equation From χQ=0 we get Solving the above two equations we get x 0 and x 1 subject to the normalizing condition x 0 e + x 1 (I − R) −1 e=1.  Φ is the absorbing state. The infinitesimal generator matrix Q 2 is given by

PERFORMANCE MEASURES
The Other important performance characteristics.
• Expected number of customers completing service without interruption E(NI)= i.x i,0 .
• Probability that there are i (i ≥ 0) customers in the system, P i =x i .e.
• Expected number of customers in the system, • Fraction of time the server in the interrupted state, FT(I)= • Fraction of time the server is busy,